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The renormalization group method in statistical hydrodynamics. (English) Zbl 0830.76042
This paper gives a first principles formulation of a renormalization group (RG) method appropriate to study of turbulence in incompressible fluids governed by Navier-Stokes equations. The present method is a momentum-shell RG of Kadanoff-Wilson type based upon the Martin-Siggia- Rose field-theory formulation of stochastic dynamics. A simple set of diagrammatic rules are developed which are exact within perturbation theory (unlike the well-known Ma-Mazenko prescriptions). It is also shown that the claim of V. Yakhot and S. A. Orszag [J. Sci. Comput. 1, 3-51 (1986; Zbl 0648.76040)] is false and that higher-order terms are irrelevant in the \(\varepsilon\) expansion \(RG\) for randomly forced Navier-Stokes equations with power-law force spectrum.

76F99 Turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] D. Forster, D. Nelson, and M. Stephen, ”Large-distance and long-time properties of a randomly stirred fluid,” Phys. Rev. A 16, 732 (1977).PLRAAN1050-2947
[2] C. DeDominicis and P. C. Martin, ”Energy spectra of certain randomly-stirred fluids,” Phys. Rev. A 19, 419 (1979).PLRAAN1050-2947
[3] J. P. Fournier and U. Frisch, ”Remarks on the renormalization group in statistical fluid mechanics,” Phys. Rev. A 28, 1000 (1983).PLRAAN1050-2947
[4] V. Yakhot and S. A. Orszag, ”Renormalization group analysis of turbulence, I. Basic theory,” J. Sci. Comp. 1, 3 (1986).JSCOEB0885-7474 · Zbl 0648.76040
[5] E. V. Teodorovich, ”Use of the renormalization-group method to describe intermittency and to derive the corrections to the exponents in Kolmogorov turbulence theory,” Sov. Phys. JETP 75, 472 (1992).SPHJAR0038-5646
[6] C. Di Castro and G. Jona-Lasinio, ”The renormalization group approach to critical phenomena,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, New York, 1976), Vol. 6.
[7] K. G. Wilson and J. B. Kogut, ”Renormalization group and the \(\epsilon\)-expansion,” Phys. Rep. C 12, 75 (1974). · doi:10.1016/0370-1573(74)90023-4
[8] K. G. Wilson, ”The renormalization group: critical phenomena and the Kondo problem,” Rev. Mod. Phys. 47, 773 (1975).RMPHAT0034-6861
[9] G. L. Eyink, ”Renormalization group and operator-product expansion in turbulence: shell models,” Phys. Rev. E 48, 1823 (1993).PLEEE81063-651X
[10] P. C. Martin, E. D. Siggia, and H. A. Rose, ”Statistical dynamics of classical systems,” Phys. Rev. A 8, 423 (1973).PLRAAN1050-2947
[11] H. K. Janssen, ”On a Lagrangian for classical field dynamics and renormalization group calculations of dynamical critical properties,” Z. Phys. B 23, 377 (1976).ZPBBDJ0340-224X
[12] C. DeDominicis, ”Techniques de renormalisation de la théorie des champs et dynamique des phénomenĕs critiques,” J. Phys. (Paris) C 1, 247 (1976).JPSOAW0022-3719
[13] S. K. Ma and G. Mazenko, ”Critical dynamics of ferromagnetics in 6-\(\epsilon\)dimensions: general discussion and detailed calculation,” Phys. Rev. B 11, 4077 (1975).PLRBAQ0556-2805
[14] E. A. Novikov, ”Functionals and the random-force method in turbulence theory,” Sov. Phys. JETP 20, 1290 (1965).SPHJAR0038-5646
[15] H. Tennekes, ”Eulerian and Lagrangian time microscales in isotropic turbulence,” J. Fluid Mech. 67, 561 (1975).JFLSA70022-1120 · Zbl 0302.76033
[16] L. Onsager, ”Reciprocal relations in irreversible processes, I, II,” Phys. Rev. 37, 405 (1931); PHRVAO0031-899X · JFM 57.1168.10
[17] L. Onsager, 38, 2265 (1931).PHRVAO0031-899X, Phys. Rev.
[18] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981).
[19] K. G. Wilson, ”Renormalization group and critical phenomena. II. phase-space cell analysis of critical behavior,” Phys. Rev. B 4, 3184 (1971).PLRBAQ0556-2805 · Zbl 1236.82016
[20] G. L. Eyink, ”Quantum field-theory models on fractal spacetime, I, II,” Commun. Math. Phys. 125, 613 (1989); CMPHAY0010-3616 · Zbl 0713.58051
[21] G. L. Eyink, 126, 85 (1989).CMPHAY0010-3616, Commun. Math. Phys.
[22] G. L. Eyink, ”Dissipation and large thermodynamic fluctuations,” J. Stat. Mech. 61, 533 (1990). · doi:10.1007/BF01027291
[23] M. Kardar, G. Parisi, and Y.-C. Zhang, ”Dynamic scaling of growing interfaces,” Phys. Rev. Lett. 56, 889 (1986).PRLTAO0031-9007 · Zbl 1101.82329
[24] J. Collins, Renormalization (Cambridge University Press, Cambridge, 1984). · Zbl 1094.53505 · doi:10.1017/CBO9780511622656
[25] L. D. Adzhemyan, A. N. Vasil’ev, and Yu. M. Pis’mak, ”Renormalization-group approach in the theory of turbulence: The dimensions of composite operators,” Teor. Mat. Fiz. 57, 268 (1983).TMFZAL0564-6162
[26] However, the same type of difficulties still occur. The formal solution to the recursion relations Eq. (53) is gk(3,s) = g0(3,s-k)+\(\Sigma\)l = 0k-1N(3,s-l)[gk-l-1] when s>k, and it can be O(1) if k/\(\epsilon\).
[27] L. M. Smith and W. C. Reynolds, ”On the Yakhot-Orszag renormalization group method for deriving turbulence statistics and models,” Phys. Fluids A 4, 364 (1992).PFADEB0899-8213
[28] V. Yakhot and L. M. Smith, ”The renormalization group, the \(\epsilon\)-expansion, and derivation of turbulence models,” J. Sci. Comput. 7, 35 (1992).JSCOEB0885-7474 · Zbl 0757.76019
[29] R. H. Kraichnan, ”Hydrodynamic turbulence and the renormalization group,” Phys. Rev. A 25, 3281 (1982).PLRAAN1050-2947
[30] M. Nelkin and M. Tabor, ”Time correlations and random sweeping in isotropic turbulence,” Phys. Fluids A 2, 81 (1990).PFADEB0899-8213 · Zbl 0696.76065
[31] C. Van Atta and J. C. Wyngarde, ”On higher-order spectra of turbulence,” J. Fluid Mech. 72, 673 (1975).JFLSA70022-1120
[32] R. Panda, E. Clemente, V. Sonnad, S. A. Orszag, and V. Yakhot, ”Turbulence in a randomly stirred fluid,” Phys. Fluids A 1, 1045 (1989).PFADEB0899-8213
[33] S. Chen and R. H. Kraichnan, ”Sweeping decorrelation in isotropic turbulence,” Phys. Fluids A 1, 2019 (1989).PFADEB0899-8213
[34] Y. Zhou, G. Vahala, and M. Hossain, ”Renormalization-group theory for the eddy viscosity in subgrid modeling,” Phys. Rev. A 37, 2590 (1988).PLRAAN1050-2947
[35] H. A. Rose, ”Eddy diffusivity, eddy noise and sub-grid scale modeling,” J. Fluid Mech. 81, 719 (1977).JFLSA70022-1120
[36] R. H. Kraichnan, ”An interpretation of the Yakhot-Orszag turbulence theory,” Phys. Fluids 30, 2400 (1987).PFLDAS0031-9171 · Zbl 0642.76069
[37] R. H. Kraichnan, ”Eddy viscosity and diffusivity: exact formulas and approximations,” Complex Systems 1, 805 (1987).CPSYEN0891-2513 · Zbl 0652.76037
[38] C. G. Speziale, ”Turbulence modeling in noninertial frames of reference,” Theor. Comp. Fluid Dyn. 1, 3 (1989).TCFDEP0935-4964 · Zbl 0712.76047
[39] A. Yoshizawa, ”A statistically derived system of equations for turbulent shear flows,” Phys. Fluids 28, 59 (1985).PFLDAS0031-9171 · Zbl 0565.76053
[40] G. L. Eyink, ”Lagrangian field theory, multifractals, and universal scaling in turbulence,” Phys. Lett. A 172, 355 (1993).PYLAAG0375-9601
[41] M. J. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydrodynamics (Kluwer Academic, Dordrecht, 1988).
[42] R. H. Kraichnan, ”Kolmogorov’s hypotheses and Eulerian turbulence theory,” Phys. Fluids 7, 1723 (1964).PFLDAS0031-9171 · Zbl 0151.41701
[43] R. H. Kraichnan, ”Langrangian-history closure approximation for turbulence,” Phys. Fluids 8, 575 (1965).PFLDAS0031-9171
[44] F. J. Wegner, ”The critical state, general aspects,” in Phase Transitions and Critical Phenomena, Vol. 6, edited by C. Domb and M. S. Green (Academic, New York, 1976).
[45] R. J. Baxter, ”Partition function of the eight-vertex lattice model,” Ann. Phys. (N.Y.) 70, 193 (1972).APNYA60003-4916
[46] G. L. Eyink, ”Energy dissipation without viscosity in ideal hydrodynamics, I. Fourier analysis and local transfer,” Physica D (to appear). · Zbl 0817.76011 · doi:10.1016/0167-2789(94)90117-1
[47] Z.-S. She (private communication).
[48] R. H. Kraichnan (private communication).
[49] R. H. Swendson, ”Monte-Carlo renormalization,” in Real-Space Renormalization, edited by W. Burkhardt and J. M. J. van Leeuwen (Springer, New York, 1982).
[50] A. M. Polyakov, ”The theory of turbulence in two dimensions,” Nucl. Phys. B 396, 367 (1993).NUPBBO0550-3213
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